Analysis of Network Reliability Characteristics and Importance of Components in a Communication Network

Network reliability is one of the most important concepts in this modern era. Reliability characteristics, component significance measures, such as the Birnbaum importance measure, critical importance measure, the risk growth factor and average risk growth factor, and network reliability stability of the communication network system have been discussed in this paper to identify the critical components in the network, and also to quantify the impact of component failures. The study also proposes an efficient algorithm to compute the reliability indices of the network. The authors explore how the universal generating function can work to solve the problems related to the network using the exponentially distributed failure rate. To illustrate the proposed algorithm, a numerical example has been taken.


Introduction
Network reliability has vital importance at all stages of processing and controlling communication networks. Apart from the reliability of the network, the component's significance is also important. The main purpose of this study is to identify the weak/critical components and quantify the impact of their failures on the network. The flow of signals transmitted from source to sink is called "terminal reliability" or network reliability [1][2][3][4][5]. Zarghami et al. [6] analyzed the exact reliability of infrastructure networks through the decomposition of the network into a set of series and parallel structures.
The universal generating function (UGF) is one of the more noteworthy methods for estimating network reliability, based on various algorithms proposed by Levitin [7]. Lisnianski and Levitin [8] used many real-world multi-state systems in which all the components had different performance levels and failure modes. Negi and Singh [9] studied the non-repairable complex system consisting of two subsystems, say, A and B, connected in series. They also evaluated the reliability, mean time to failure (MTTF), and sensitivity of the considered system with the use of UGF. Renu et al. [10] analyzed the reliability of repairable parallel-series multi-state systems by the application of interval UGF. The authors computed the probabilities of different components, reliability, sensitivity, and MTTF with the use of the Markov process and the Laplace-Steiltjes transform method. The Markov and supplementary variable technique. Ram and Singh [37] studied the reliability characteristics of a complex system using the Markov process and Gumbel-Houggard family of copula. Ram [38] studied the reliability analysis of various engineering systems using copula and Markov process techniques.
From the above discussions, it is clear that, previously, several researchers had calculated the reliability of different types of networks using the minimal cuts and path methods from a probabilistic approach, such as inclusion-exclusion, product disjoints, and factoring methods. In this paper, the authors discuss the reliability characteristics of a communication network with respect to the different parameters and also propose an algorithm to find the reliability function of the considered network. Numerical examples have been taken to discuss the findings of the communication network as shown in Figure 1, in which different edges have different exponentially distributed failure rates. The notations used in the proposed network have been listed in Table 1.
reliability analysis with the application of an artificial neural network approach [34]. Ram and Manglik [35,36] analyzed the reliability model of an industrial system having three subsystems, using the Markov and supplementary variable technique. Ram and Singh [37] studied the reliability characteristics of a complex system using the Markov process and Gumbel-Houggard family of copula. Ram [38] studied the reliability analysis of various engineering systems using copula and Markov process techniques.
From the above discussions, it is clear that, previously, several researchers had calculated the reliability of different types of networks using the minimal cuts and path methods from a probabilistic approach, such as inclusion-exclusion, product disjoints, and factoring methods. In this paper, the authors discuss the reliability characteristics of a communication network with respect to the different parameters and also propose an algorithm to find the reliability function of the considered network. Numerical examples have been taken to discuss the findings of the communication network as shown in Figure  1, in which different edges have different exponentially distributed failure rates. The notations used in the proposed network have been listed in Table 1.
The rest of this paper is organized as follows: estimation of network reliability using UGF, and algorithms for drawing the reliability function, model description, and numerical illustration of the network are introduced in Section 2. In Section 3, the MTTF of the network with respect to each failure rate has been computed. In Section 4, the authors evaluated the importance of components in the networks from the five metrics. Finally, the conclusions are drawn in Section 5.   Probability of the set of nodes ψ n:m does not receive a signal from a node located at ω ω-function operator R 5 Reliability of communication network.  Probability of x which is equal to x m r n nth node of the considered network ψ n:m Set of nodes receive a signal from the node located at r n p m,ψ n:m Probability of the set of node ψ n:m receiving a signal directly from node situated at r n q m,ψ n:m Probability of the set of nodes ψ n:m does not receive a signal from a node located at r n ω ω-function operator R 5 Reliability of communication network.
The rest of this paper is organized as follows: estimation of network reliability using UGF, and algorithms for drawing the reliability function, model description, and numerical illustration of the network are introduced in Section 2. In Section 3, the MTTF of the network with respect to each failure rate has been computed. In Section 4, the authors evaluated the importance of components in the networks from the five metrics. Finally, the conclusions are drawn in Section 5.

Network Description
The study of networks is a vast field that deals with the real-life applications of networks in communication, software engineering, industry, mobile ad-hoc, etc. A network is a combination of nodes interconnected with edges. The model of a network is defined by G = (N, E) in which "N" and "E" are the nodes and edges, respectively [2]. In communication networks, the nodes represent computers and the edges, in which the information is transmitted through various transmission lines, connect these computers.
A proposed network consists of a root node where the signal source is placed, and several intermediate nodes receive a signal, which is capable of transmitting the messages to the sink node. The proposed network is used as the communication network.

Universal Generating Function
The UGF method was first discussed by Ushakov [39] to find the reliability of systems or networks. Levitin [7] presents a detailed description of UGF, composition operator, and network reliability.
A polynomial defines the UGF of a discrete random variable as: where x has m probable values and p m is the probability of x, which is equal to x m .

Estimation of Network Reliability Using the Universal Generating Function
Assume a node is placed at position r n . If in-state m (1 ≤ m ≤ M n ) is the node available for signal transmission from r n to a set of nodes ψ n:m , then it is represented by 1, otherwise by 0: v n:m = 1, r n ∈ ψ n:m 0, r n / ∈ ψ n:m The node-UGF of v n elements are expressed as: where ω is introduced over u i (z) and u i+1 (z), the subnet-UGF of U i+1 (z) elements are expressed as: • From the operator ω, remove the term from UGF where the path does not go through the considered node (unit), and also if the path does not complete from the source node to the considered node. • For various nodes, collect all similar terms in the resulting UGF.

Algorithm for Determining the Reliability of Networks
For a binary state network (communication network), an algorithm is developed to evaluate network reliability [14] as follows: Step 1: Find out vectors v n:m corresponding to sets ψ n:m for the nodes located at the positions r 1 , . . . . . . , r L−M in the network.
Step 4: Step 5: Simplify polynomial U L−M (z), then, using operator ω, obtain the network reliability at the sink (terminal) nodes.
Here, we have taken two different networks as a case study to examine the reliability characteristics.

Model Description
The possibilities and conditions for moving signal flows are as follows.

Numerical Illustration Reliability Computation of Communication Network
Consider a communication network as shown in Figure 1, when the flow of signal originates from node 1 and terminates at node 6, the considered network having total node L = 6 and the sink node M = 1, when there exists a node for each subset of ∧ n (1 ≤ n ≤ 5).
The UGF of the nodes, node 1, node 2, node 3, node 4, and node 5 are expressed as: Using operator ω, the number of terms in UGF are decreased from 6 to 3. The UGF of node 3 is given by: Here, using the operator ω, the number of terms in the UGF is decreased from 3 to 1. ω[U 5 (z)] is the probability that the signal reaches to sink node 6 via node 5, which yields the network reliability.
The variation of reliability with respect to the time of the proposed network has been obtained from Equation (6). The different values of reliability obtained with respect to time are shown in Table 2, and the corresponding graph has been depicted in Figure 2.

Mean Time to Failure (MTTF)
MTTF is defined as the consecutive failures before some sources are disconnected from the destination. It is the mean time of the network until the first failure occurs, and it is related to reliability [9,10].

Mean Time to Failure (MTTF)
MTTF is defined as the consecutive failures before some sources are disconnected from the destination. It is the mean time of the network until the first failure occurs, and it is related to reliability [9,10].
MTTF with respect to the failure rate for different components of the network, i.e., 1: From the above equation, finding the MTTF for the different edges, increasing the value of failure rate with the corresponding parameters λ 12 , and setting all other failure rates as constant, one can obtain variation in MTTF with respect to λ 12 from Equation (7); values are given in Table 3, and their corresponding graph in Figure 3. Similarly, for the parameters λ 23 , λ 24 ,λ 35 , λ 45 , λ 56 and λ 13 , we can obtain variation in MTTF from Equation (7), and all values are given in Table 3, and their corresponding graph in Figure 3.  13 .
From the above equation, finding the MTTF for the different edges, increasing the value of failure rate with the corresponding parameters λ 12 , and setting all other failure rates as constant, one can obtain variation in MTTF with respect to λ 12 from Equation (7); values are given in Table 3, and their corresponding graph in Figure 3. Similarly, for the parameters λ 23 , λ 24 ,λ 35 , λ 45 , λ 56 and λ 13 , we can obtain variation in MTTF from Equation (7), and all values are given in Table 3, and their corresponding graph in Figure 3.

Birnbaum Component Importance
With the development of modern technology, network systems are becoming increasingly complex nowadays. Reliability engineers need a mathematical approach for the complex networks, which can provide the means to define the Birnbaum component importance [21]. The Birnbaum measure [21], denoting the importance of the network adopted by the reliability of the network and the component, is presented in the network:

Birnbaum Component Importance
With the development of modern technology, network systems are becoming increasingly complex nowadays. Reliability engineers need a mathematical approach for the complex networks, which can provide the means to define the Birnbaum component importance [21]. The Birnbaum measure [21], denoting the importance of the network adopted by the reliability of the network and the component, is presented in the network: